Periodic motion

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Periodic phenomena or event repeats after certain interval like seasons, motion of planets, swings, pendulum etc. In this module, our focus, however, is the physical movement of particle or body, which shows a pattern recurring after certain fixed time interval.

Representation of periodic motion has a basic pattern, which is repeated at regular intervals. What it means that if we know the basic form (building block), then we can describe the motion by following the pattern again and again.

Periodic attributes

A periodic motion can be described with respect to different quantities. A given periodic motion can have a host of attributes which may undergo periodic variations. Consider, for example, the case of a pendulum. We can choose any of the attributes like angle (θ), horizontal displacement (x), vertical displacement (y), kinetic energy (K), potential energy (U) etc. The values of these quantities undergo periodic alteration with respect to time. These attributes constitute periodic attributes of the periodic motion.

There are, however, other attributes, which may remain constant during periodic motion. If we consider pendulum executing simple harmonic motion (it is a particular periodic motion), then total mechanical energy of the system is constant and as such is independent of time. Hence, we need to pick appropriate attribute(s) to describe a periodic motion in accordance with problem situation in hand.

Description of periodic motion

We need a mathematical model to describe periodic motion. For this, we employ certain mathematical functions. The important feature of a periodic function is that its value is repeated after certain interval. We call this interval as “period”. In case, this period refers to time, then the same is called “time period”. Mathematically, a periodic function is defined as :

Periodic motion
A function is said to be periodic if there exists a positive real number “T”, which is independent of “t”, such that $f\left(t+T\right)=f\left(t\right)$ .

The least positive real number “T” (T>0) is known as the fundamental period or simply the period of the function. The “T” is not a unique positive number. All integral multiple of “T” is also the period of the function.

In the context of periodic function, an “aperiodic” function is one, which in not periodic. On the other hand, a function is said to be anti-periodic if :

$f\left(t+T\right)=-f\left(t\right)$

Example

Problem 1: Prove that " $\mathrm{sin}t$ " is a periodic function. Also find its period.

Solution :

We know that :

$\mathrm{sin}\left(n\pi +t\right)={\left(-1\right)}^{n}\mathrm{sin}t,\phantom{\rule{1em}{0ex}}\text{when “n” is an integer.}$

$\mathrm{sin}\left(n\pi +t\right)=\mathrm{sin}t,\phantom{\rule{1em}{0ex}}\text{when “n” is an even integer.}$

Thus, there exists T>0 such that f(t+T) = f(t). Further “nπ” is independent of “t”. Hence, “ $\mathrm{sin}t$ ” is a periodic function. Its period is the least value, when n = 2 (first even positive integer),

$⇒T=2\pi$

The period of "sine" function is collaborated from the figure also. Note that we can not build a sine curve with a half cycle of period “π” (upper figure). We require a full cycle of “2π” to build a sine curve (lower figure).

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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